random matrices and random partitions normal convergence world scientific series on probability theory and its applications

Download Book Random Matrices And Random Partitions Normal Convergence World Scientific Series On Probability Theory And Its Applications in PDF format. You can Read Online Random Matrices And Random Partitions Normal Convergence World Scientific Series On Probability Theory And Its Applications here in PDF, EPUB, Mobi or Docx formats.

This book is aimed at graduate students and researchers who are interested in the probability limit theory of random matrices and random partitions. It mainly consists of three parts. Part I is a brief review of classical central limit theorems for sums of independent random variables, martingale differences sequences and Markov chains, etc. These classical theorems are frequently used in the study of random matrices and random partitions. Part II concentrates on the asymptotic distribution theory of Circular Unitary Ensemble and Gaussian Unitary Ensemble, which are prototypes of random matrix theory. It turns out that the classical central limit theorems and methods are applicable in describing asymptotic distributions of various eigenvalue statistics. This is attributed to the nice algebraic structures of models. This part also studies the Circular β Ensembles and Hermitian β Ensembles. Part III is devoted to the study of random uniform and Plancherel partitions. There is a surprising similarity between random matrices and random integer partitions from the viewpoint of asymptotic distribution theory, though it is difficult to find any direct link between the two finite models. A remarkable point is the conditioning argument in each model. Through enlarging the probability space, we run into independent geometric random variables as well as determinantal point processes with discrete Bessel kernels. This book treats only second-order normal fluctuations for primary random variables from two classes of special random models. It is written in a clear, concise and pedagogical way. It may be read as an introductory text to further study probability theory of general random matrices, random partitions and even random point processes. Contents:Normal ConvergenceCircular Unitary EnsembleGaussian Unitary EnsembleRandom Uniform PartitionsRandom Plancherel Partitions Readership: Graduates and researchers majoring in probability theory and mathematical statistics, especially for those working on Probability Limit Theory. Key Features:The book treats only two special models of random matrices, that is, Circular and Gaussian Unitary Ensembles, and the focus is on second-order fluctuations of primary eigenvalue statistics. So all theorems and propositions can be stated and proved in a clear and concise languageIn a companion part, the book also treats two special models of random integerpartitions, namely, random uniform and Plancherel partitions. It exhibits a surprising similarity between random matrices and random partitions from the viewpoint of asymptotic distribution theory, though there is no direct link between finite modelsThe limit distributions of most statistics of interest are obtained by reducing to classical central limit theorems for sums of independent random variables, martingale sequences and Markov chains. So the book is easily accessible to readers that are familiar with a standard probability theory textbookKeywords:Central Limit Theorems;Random Matrices;Random Partitions

This book is an excellent introduction to probability theory for students who have some general experience from university-level mathematics, in particular, analysis. It would be suitable for reading in conjunction with a second or third year course in probability theory. Besides the standard material, the author has included sections on special topics, for example percolation and statistical mechanics, which are direct applications of the theory.

The book is suitable for a lecture course on the theory of Brownian motion, being based on final year undergraduate lectures given at Trinity College, Dublin. Topics that are discussed include: white noise; the Chapman-Kolmogorov equation — Kramers-Moyal expansion; the Langevin equation; the Fokker-Planck equation; Brownian motion of a free particle; spectral density and the Wiener-Khintchin theorem — Brownian motion in a potential application to the Josephson effect, ring laser gyro; Brownian motion in two dimensions; harmonic oscillators; itinerant oscillators; linear response theory; rotational Brownian motion; application to loss processes in dielectric and ferrofluids; superparamagnetism and nonlinear relaxation processes. As the first elementary book on the Langevin equation approach to Brownian motion, this volume attempts to fill in all the missing details which students find particularly hard to comprehend from the fundamental papers contained in the Dover reprint — Selected Papers on Noise and Stochastic Processes, ed. N Wax (1954) — together with modern applications particularly to relaxation in ferrofluids and polar dielectrics. Contents:Historical Background and Introductory ConceptsLangevin Equations and Methods of SolutionThe Brownian Motion of a Free Particle and a Harmonic OscillatorThe Itinerant Oscillator ModelTwo-Dimensional Rotational Brownian Motion in N-Fold Cosine PotentialsThe Brownian Motion in a Tilted Cosine Potential: Application to the Josephson Tunnelling JunctionThree-Dimensional Rotational Brownian Motion in an External Potential with Application to the Theory of Dielectric and Magnetic RelaxationRotational Brownian Motion in an External Potential — Matrix Continued Fraction SolutionNumerical Solutions for Non-Axially Symmetric ProblemsInertial Langevin Equations: Application to the Theory of Dielectric and Kerr-Effect RelaxationLinear Response Theory and the Fokker-Planck Operator Readership: Physicists, chemists, electrical engineers, statisticians and undergraduates. keywords:Langevin Equation;FokkerâPlanck Equation;relaxation and Stochastic Processes;Rotational Diffusion;Diffusion in a Potential;Kramers' Theory;Linear and Nonlinear Response Theory;Dielectric Relaxation;Superparamagnetism;Josephson Effect “I found this book a valuable addition to my library. It will be of interest to researchers and advanced students and the material could be used as the text for a course for advanced undergraduates and graduate students.” Journal of Statistical Physics

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